Download PPT - Fernando Brandao

A reversible framework for entanglement and
other resource theories
Fernando G.S.L. Brandão
Universidade Federal de Minas Gerais, Brazil
Martin B. Plenio
Ulm University, Germany
Enlight 2010
What this talk is about:
1.
An analogy of entanglement theory and the second law of
thermodynamics
2.
Using the analogy to solve open problems in entanglement
theory
3.
Extending the analogy to almost every resource theory
4.
Implications to Bell inequalities and its statistical
significance
What this talk is not about:
1.
2.
3.
Drawing work from entangled states
Thermal entanglement in equilibrium systems
ETC….
In particular, no knowledge of real
thermodynamics is assumed or needed…
Quantum Entanglement
Quantum Entanglement
i
p



 i A B
i
i
Quantum Entanglement
   pi   ...  
i
1
i
k
i
Entanglement is
What cannot be created by local operations and classical
communication (LOCC)
Restricted Operations and Resources

In Physics we commonly deal with restrictions on the physical
processes/operations available. Entanglement theory is an
instance of such paradigm.
Restricted set
of operations

Local operations
and classical
communication
(LOCC)
Free states
Resource
Separable states
Entanglement
The Holy Grail of Resource Theories

Given some set of allowed operations on a
physical system, which transformations
from one state of the system into another
can be realized?
When and how can we transform a
resource from one form into another?
LOCC transformations

• Given two quantum states, can
we transform one into the other
by LOCC?
A
B
• At what rate can we make the
transformation?
• We will be interested in this
question in the limit of a large
number of copies of the two
states


n
 LOCC  n  
 kn
LOCC asymptotic entanglement transformations

Optimal rate of conversion:
A
B
How many copies of rho we
have to invest per copy of
sigma?

Mathematically:
n

n
m


R(    )  inf  : lim  min || (  )   ||1   0

 m n LOCC

Bipartite pure state entanglement transformations
• The simplest form of entanglement: 
AB
  k k
A
k
k
• (Bennett et al 96) Transformations are reversible

n
 LOCC 
 nE ( ) / E ( )
 LOCC 
n
with E(  )  S (  A ) the unique measure of entanglement
• Equivalent to the existence of a total order

n
 LOCC 
n
if, and only if,
E ( )  E ( )
B
Mixed state entanglement
• Entanglement cost:
EC (  )  R(2   )
2  (| 00   |11 ) / 2

Mixed state entanglement
• Distillable entanglement:
2  (| 00   |11 ) / 2
ED (  )  R(   2 )

1
Mixed state entanglement
Manipulation of mixed state entanglement under LOCC is
irreversible
• (3 x Horodecki 98) In general:
EC (  )  ED (  )
• Extreme case, bound entanglement:
EC (  )  0, ED (  )  0
• No unique measure for entanglement manipulation under
LOCC. In fact there is a whole zoo of them...
Thermodynamics and its second law
• (Clausius, Kelvin, Planck, Caratheodory) The second law of
thermodynamics: The entropy of a system in thermal
equilibrium can never decrease during an adiabatic process (a
transformation which doesn’t involve exchange of heat with
environment)
• In 1999 Lieb and Yngvason proposed an axiomatic approach to
the second law, which try to improve on the non-rigorous
arguments of the previous formulations.
•It’s in the mathematical structure they identified which we are
most interested!
Lieb and Yngvason Approach to 2nd Law
 X, Y, Z are thermodynamical states
X
Y : there is an adiabatic transformation from X to Y
 (X, Y): composition of the two systems X and Y
 t X: system composed of t times X.

Axioms:
1.X
X
2.If X
Y and Y
Z then X
Z
3.X
Y and X’
Y’ then (X, X’)
(Y, Y’)
4.X
Y then tX
tY for all t
5.X
(t X, (1-t)X) for all 0 < t < 1
6.If (X, s Z)
(Y, s Z) for s
0 then X
Y
Comparison Hypothesis:
For all X, Y either X
Y or Y
X
Lieb and Yngvason Formulation of the 2nd Law
(Lieb and Yngavason 99) Axioms 1-6 and the Comparison
Hypothesis imply that there is a function S on the state
space such that
X
Y if, and only if, S(X) <= S(Y)
The function S can be taken to be additive, i.e.
S( (t X, s Y) ) = t S(X) + s S(Y)
and then it is unique up to affine transformations
Entanglement Theory and the Second Law
• The LY formulation of the 2nd law is formally equivalent to
the law of transformations of bipartite pure states by LOCC
• All LY axioms are true for bipartite pure states
• But for general (mixed) states, the analogy breaks down:
there is bound entanglement...
Entanglement Theory and the Second Law
• Work by Horodecki&Oppenheim, Popescu&Rorhlich, Plenio&Vedral,
Vedral&Kashefi explored this connection through several
angles and asked whether one could somehow find a
thermodynamical formulation for general states
• BTW: Thinking about this had already led to technical
progress in entanglement theory: The Horodeckis
discovered entanglement activation inspired by the
thermodynamical analogy...
Entanglement in Fantasy Land
• We will look at a thermodynamical formulation
for entanglement theory by extending the class
of operations allowed beyond LOCC
• Extending the class of operations to get a
more tractable theory is not a new idea (Rains,
Eggeling et al, Audenaert et al, Horodecki et al):
1. Separable operations
2. PPT operations
3. LOCC + bound entanglement
Non-entangling maps
• We will consider the extreme situation, and consider the
manipulation of entanglement by all transformations which
don’t generate entanglement
• The class of non-entangling maps consists of all quantum
operations which maps separable states to separable states
Reversibility under Non-entangling Maps
• (B, Plenio 08: Informal) Under the class of (asymptotic) nonentangling operations, entanglement theory is reversible.
There is an entanglement measure E such that for every all
states rho, sigma,

n

 nE (  ) / E ( )

n
and, equivalently,
 n   n
if, and only if,
E (  )  E ( )
Two measures of entanglement
• Relative entropy of entanglement:
ER (  )  min S (  ||  )
 S
S (  ||  )  tr(  (log   log  ))
Vedral&Plenio 97
• Robustness of Entanglement:
  s
rG (  )  min s :
S
1 s
Vidal&Tarrach 99 and Harrow&Nielsen 03
Asymptotically non-entangling operations
 is  -non-entangling if
rG (( ))  
• Definition: A quantum operation
For every separable state

• We say that a sequence of maps { n } is asymptotically non-
entangling if each  n is
 n -non-entangling and
lim  n  0
n 
Optimal rate of conversion
• Optimal rate of conversion under asymptotically non-
entangling operations:
n



n
m
R(    )  inf  : lim  min || (  )   ||1   0, lim  n  0
n 

 m n NE ( n )

•
NE ( )
denotes the class of
 - resource non-generating operations
Reversibility Again
For every quantum states  , 
R(    )  ER ( ) / ER (  )
n
E
(

)

R
ER (  )  lim
n 
n
Implies:
 n   n iff ER (  )  ER ( )
Applications
• Our result has many applications to the LOCC manipulation of
entanglement:
1. (B., Plenio 09) Proof that
all entangled sigma
RLOCC (    )
is positive for
2. (Beigi, Shor 09) Proof that the set of PPT states give no
useful approximation to the set of separable states
3. (B. 10) Proof that no finite set of three qubit states forms a
Minimum-Reversible-Entanglement-Generating-Set
4. (B., Horodecki 10) Progress characterizing all mixed states
which are reversible: they all consist of ensembles
of pure states perfectly distinguishable by separable
measurements.
Connection to LY approach
• The structure form of our theorem is the same as LY
formulation to the second Law. However, are all of his axioms
true?
Connection to LY approach
• The structure form of our theorem is the same as LY
formulation to the second Law. However, are all of his axioms
true?
• The swap and the identity are non-entanling, but their tensor
product is not! Not directly a counterexample, but...
Connection to LY approach
• The potential failure of axiom 3 is equivalent to the non-
additivity of the unique entanglement measure in the theory
• Open question: Is
E ( )

R
additive?
• Interesting from the viewpoint of understanding the necessity
of LY axioms
• Would also have consequences to quantum complexity theory:
(B., Horodecki 10) Additivity implies that QMA(k) = QMA(2) for all
k>2
Is entanglement special?
• The result can actually be extended far more broadly than to
entanglement. Consider a general resource theory:
Restricted set
of operations
Free states
Resource
• Let M be the set of free states in the theory of M-resources
• Assume M is closed and convex (the theory allows for mixing)
Two measures of resource
• The relative entropy of M-resource is given by
EM (  )  min S (  ||  )
 M
S (  ||  )  tr(  (log   log  ))
Vedral&Plenio 97
•The robustness of M-resource is given by
  s
rM (  )  min s :
M
1 s
Vidal&Tarrach 99 and Harrow&Nielsen 03
Asymptotically resource non-generating operations
• Definition: A quantum operation  is
rM (( ))  
 -resource non-generating if
for every non-resource state 
• We say that a sequence of maps { n } is asymptotically resource
non-generating if each
n
is
 n-resource non-generating and
lim  n  0
n 
Reversibility for Resource Theories
Under a few assumptions on M, for every quantum states
R(    )  EM ( ) / EM (  )
n
E
(

)

M
EM (  )  lim
n 
n
Implies:
 n   n iff EM (  )  EM ( )
 ,
The main idea
 The main idea is to connect the convertibility of resource
states to the distinguishability of resource states from nonresource ones
 Basically, if a resource theory is such that we can distinguish,
by measurements, many copies of a resource state from nonresource states pretty well, then the theory is reversible under
the class of resource non-generating operations!
The main idea
• Quantum Hypothesis Testing: given several copies of a quantum
state and the promise that you are given either
(alternative hypothesis), decide which you have

(null hypothesis) or

The main idea
• Quantum Hypothesis Testing: given several copies of a quantum
state and the promise that you are given either
(alternative hypothesis), decide which you have
• Quantum Stein’s Lemma:
1 ( An ) : tr(   n ( I  An ))
 2 ( An ) : tr (  n An )
Hiai&Petz 91 and Ogawa&Nagaoka 99

(null hypothesis) or

The main idea
• Resource Hypothesis Testing: given a sequence of quantum states
H  n , with the promise that either n n is a sequence
n
of unknown non-resource states or  n  
, for some resource state 
n
acting on
decide which is the case.
Probabilities of Error:
1 ( An ) : tr(  ( I  An ))
n
 2 ( An ) : max tr(n An )
n M
,
The main idea
• We say a resource theory defined by the non-resource states M has the
exponential distinguishing (ED) property if for every resource state
 n (  ,  ) : min  2 ( An ) : 1 ( An )     2

 nE (  )
0 An  I
For a non-identically zero function E
 2 ( An ) : max tr(n An )
n M
1 ( An ) : tr(  ( I  An ))
n
Theorem I
• Theorem: If M satisfies ED, then
log( 1  rM (  n ))
E (  )  min lim
: ||  n   n ||1  0
{  n } n
n
and for

such that
E ( )  0
R(    )  E (  ) / E ( )
Theorem I
• Theorem: If M satisfies ED, then
log( 1  rM (  n ))
E (  )  min lim
: ||  n   n ||1  0
{  n } n
n
and for every

such that
E ( )  0
R(    )  E (  ) / E ( )
Theorem I
• Theorem: If M satisfies ED, then
log( 1  rM (  n ))
E (  )  min lim
: ||  n   n ||1  0
{  n } n
n
and for every

such that
E ( )  0
R(    )  E (  ) / E ( )
Proof main idea: Take maps of the form
n (.)  tr( An .) nE( ) / E (  )  tr(( I  An ).) n
Theorem I
• Theorem: If M satisfies ED, then
log( 1  rM (  n ))
E (  )  min lim
: ||  n   n ||1  0
{  n } n
n
and for every

such that
E ( )  0
R(    )  E (  ) / E ( )
The theorem is completely general.
The trouble is of course how to prove that the set of non-resource
of interest satisfies ED... Difficult in general!
Theorem II
• Theorem: If M satisfies
1. Closed and convex and contain the max. mixed state
2. If
3. If
4. If
  M n ,   M m      M nm
  M n1  trn1 ( )  M n
  M n  P P  M n ,   Sn
Theorem II
• Theorem: If M satisfies
1. Closed and convex and contain the max. mixed state
2. If
3. If
4. If
  M n ,   M m      M nm
  M n1  trn1 ( )  M n
  M n  P P  M n ,   Sn
P  1  2 ...  n    1 (1)   1 ( 2) ...   1 ( n)
Theorem II
• Theorem: If M satisfies
1. Closed and convex and contain the max. mixed state
2. If
3. If
4. If
  M n ,   M m      M nm
  M n1  trn1 ( )  M n
  M n  P P  M n ,   Sn
Then ED holds true,
E (  )  EM (  ) ,
and by the Theorem I
R(    )  E ( ) / E (  )

M

M
Theorem II
• Theorem: If M satisfies
1. Closed and convex and contain the max. mixed state
2. If
3. If
4. If
  M n ,   M m      M nm
  M n1  trn1 ( )  M n
  M n  P P  M n ,   Sn
Then ED holds true,
E (  )  EM (  ) ,
and by the Theorem I
R(    )  E ( ) / E (  )

M

M
Proof: Original quantum Stein’s Lemma + exponential de Finetti theorem +
Lagrange duality
Resource Theories
• The theorem applies to many resource theories (purity,
non-classicality, etc). A notable example where it fails is in
resource theories of superselection rules:
In this case, the theories do not satisfy ED (Gour, Marvian,
Spekkens 09)
• There is also a notable example where the formalism works
well: Non-locality
Non Locality
Alice
x=1,…,m
y=1,…,m
a=1,…,r
b=1,…,r
Bob
 Alice and Bob chooses uniformly at random from m measurements,
each with r outcomes, sampling from the joint distribution
p(a, b x, y)
 Classically, all the possible distributions have the form
p(a, b x, y)   p  pa x,  qb y,  
i.e. they all admit a local hidden variable theory
Non Locality
Alice
x=1,…,m
y=1,…,m
a=1,…,r
b=1,…,r
Bob
 Quantum mechanics allow more general distributions! Bell inequalities
provide an way to experimentally probe this.
 Our result on hypothesis testing gives a statistically stronger “Bell test”.
Non-Locality Hypothesis Testing
Alice
x1, x2, .., xn
y1, y2,.., yn
a1, a2, .., an
b1,b2,..,bn
Bob
• Given a sequence of random variables n  (( x1, a1 , y1 , b1 ),..., ( xn , an , yn , bn )
with the promise that either they are distributed according to
p(a, b x, y)
or they are distributed according to classical distributions (admitting a LHV),
but which might be arbitrary and vary for each n
Let
1 ( An ),  2 ( An ) be the probabilities of error for the test A
n
,
Non-Locality Hypothesis Testing
• Then we have:
 n ( p,  ) : min  2 ( An ) : 1 ( An )     2
0 An 1
with
ENL ( p(a, b | x, y)) : min S ( p || q)
qLHV

 nENL
( p)
Non-Locality Reversibility
• The class of operations here as just stochastic matrices
mapping LHV distributions to LHV distributions. So reversibility
means
• All non-local probability distributions are qualitatively the same:
n samples of an arbitrary distribution p(a, b | x, y) has the
same statistical strength as nE

NL

NL
( p) / E ( pPR ) samples


of a Popescu-Rohrlich box or nE NL ( p ) / E NL ( q ) samples
of any other distribution q!
• ENL is therefore a good measure of non-locality!
Conclusions
• We have seen that it is possible to formulate a (fantasyland) thermodynamical theory of entanglement and other
resource theories
• This relaxed theory turns out to be useful for the standard
paradigm of entanglement theory
• The connection of distinguishability and reversibility, at
the heart of the proof of our result, has also implications to
non-locality tests
Thank you!!